Martin Winter - Edge-Transitive Polytopes

It has long been known that there are five regular polyhedra (the Platonic solids), six regular 4-polytopes and exactly three regular d-polytopes for all d>4. Hence, the symmetry requirement of regularity (aka. flag-transitivity) appears to be quite restrictive for convex polytopes. In contrast, the class of vertex-transitive polytopes (all vertices are identical under the symmetries of the polytope) is almost as rich as the finite groups.
In this talk we ask about the class of edge-transitive (convex) polytopes, that is, polytopes in which all edges are identical under the symmetries of the polytope. Despite this restriction feeling more similar to vertex-transitivity than to regularity, we will see that the contrary seems to be the case: the class of edge-transitive polytopes appears to be quite restricted. We give, what we believe to be, a complete list of all edge-transitive polytopes, as well as a full classification for certain interesting sub-classes. Thereby, we show how edge-transitive polytopes can be studied with the tools of spectral graph theory.